Indeed we conjecture that in any analytic normed plane a completely regular bar-joint framework is globally rigid if and only if it is 2-connected and redundantly rigid. Note that it is relatively straightforward to deduce that 2-connectivity is a necessary condition and the fact that \(H\)  is globally rigid (Theorem REF ) shows that 3-connectivity is not necessary. However it is a substantial task to show that every 2-connected and redundantly rigid graph is globally rigid in an analytic normed plane. Note that, by Theorem REF  and [1]}, the combinatorial conditions of redundant rigidity and 2-connectivity are exactly the same as those that arise in the study of global rigidity for frameworks on the cylinder in \(\mathbb {R}^3\)  [2]}. This gives additional motivation for our small graph results since \(K_5^-\)  and \(H\)  are exactly the base graphs used in the recursive proof of that work.
